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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V6">right_unital</a> <a href="vectsp_1.html#V7">distributive</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E2:60"/><i><font color="Green">E43</font></i>: 
 <a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E3:60"/><i><font color="Green">E44</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:60"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="polynom5.html#K3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="polynom5.html#K2" target="_self">`^</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom5.html#D5" target="_self">Def5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1"><b>suppose </b></a><a NAME="E1:60_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:60_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E3:60_1_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:60_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:60_1_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E2:60_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><a NAME="E3:60_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="polynom5.html#E1:60_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:60_1_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="polynom5.html#E2:60_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E5:60_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:60_1_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:60_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="polynom5.html#E4:60_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><b>then </b><i><font color="Green">E51</font></i>: <font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="polynom5.html#E6:60_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/><b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="polynom5.html#E1:60_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="polynom5.html#K3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="polynom5.html#K2" target="_self">`^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E4:60"><i><font color="Green">E45</font></i></a>, <a class="txt" href="polynom5.html#E1:60_1_1_1"><i><font color="Green">E47</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="polynom5.html#K3" target="_self">*</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E8:60_1_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="polynom5.html#T21" target="_self">Th21</a></i>
<br/>.= 
 <a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="polynom5.html#T29" target="_self">Th29</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>
;<br/></div><b>end;</b></div><b>hence </b> <a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="polynom5.html#E2:60"><i><font color="Green">E43</font></i></a>, <a class="txt" href="polynom5.html#E4:60_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="polynom2.html#T5">POLYNOM2:5</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="polynom5.html#E4:60_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="polynom5.html#K3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="polynom5.html#K2" target="_self">`^</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E4:60"><i><font color="Green">E45</font></i></a>, <a class="txt" href="polynom5.html#E4:60_1_1"><i><font color="Green">E46</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="polynom5.html#K3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="polynom5.html#K2" target="_self">`^</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="polynom5.html#K3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="polynom5.html#K2" target="_self">`^</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="polynom5.html#K3" target="_self">*</a> <span class="p1">(<span class="default"><a href="polynom3.html#K13">1_.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span><a href="algseq_1.html#K5">%&gt;</a></span>
<b>by </b><i><a class="ref" href="polynom5.html#T30" target="_self">Th30</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2"><b>suppose </b></a><a NAME="E1:60_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:60_1_2"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="polynom5.html#E3:60"><i><font color="Green">E44</font></i></a>, <a class="ref" href="polynom4.html#T8">POLYNOM4:8</a></i>;<br/><b>hence </b> <a href="polynom5.html#K5" target="_self">Subst</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
 <a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="polynom5.html#T50" target="_self">Th50</a></i>
<br/>.= 
<span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="algseq_1.html#K5">%&gt;</a></span>
<b>by </b><i><a class="ref" href="polynom5.html#T35" target="_self">Th35</a></i>
<br/>.= 
<span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> 0</span>)</span></span><a href="algseq_1.html#K5">%&gt;</a></span>
<b>by </b><i><a class="txt" href="polynom5.html#E2:60_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

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